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J. Phys. Chem. A 2010, 114, 13222–13227
Isomerization of Bicyclo[1.1.0]butane by Means of the Diffusion Quantum Monte Carlo Method Raphael Berner and Arne Lu¨chow* Institute of Physical Chemistry, RWTH Aachen UniVersity, 52056 Aachen, Germany ReceiVed: September 9, 2010; ReVised Manuscript ReceiVed: NoVember 9, 2010
The isomerization of bicyclo[1.1.0]butane which comprises species with different multireference character is studied by means of the diffusion quantum Monte Carlo method (DMC). Accurate multireference DMC calculations are presented. It can be shown that at most three configuration state functions are required to achieve a balanced description of dynamical and nondynamical electron correlation. A general scheme is described that promotes efficient error cancellation in multireference DMC calculations. Introduction In the late 1960s the first experiments on the thermal pericyclic rearrangement of bicyclo[1.1.0]butane (bicybut) to buta-1,3-diene (t-but) were conducted.1-4 The chemical relevance resides in the fact that it is a prototype for thermal reactions of bicyclobutane derivates. The present reaction with all relevant transition states and intermediate products is depicted in Scheme 1. The ring opening comprises the breaking of two σ-bonds yielding a gauche form of 1,3-butadiene (g-but) as an intermediate product. This is achieved by a rotation of the methylene groups which can proceed via a disrotatoric or a conrotatoric manner. The corresponding transition states are referred to as con_TS and dis_TS. According to the Woodward-Hoffmann rules5 the isomerization runs through the conrotatoric pathway which has been confirmed by experimentalists.2,3 g-but is converted to trans-1,3-butadiene by a rotation around the σ-bond. The corresponding transition state is referred to as gt-TS in this paper. Experiments revealed an excitation energy of 40.6 ( 2.5 kcal/ mol.1 This is in very good agreement with multireference configuration interaction (MRCI)6 and second-order multireference perturbation theory (MRPT)6 calculations, performed in the following decades which yielded an activation barrier of 41.5 kcal/mol for the concerted conrotatory pathway. Additionally they revealed that the orbital symmetry forbidden disrotatory path (dis_TS) has a transition state (TS) located about 15 kcal/ mol above the TS of the conrotatory path (con_TS).6 This reaction path poses a great challenge for many quantum chemical methods. This is mainly due to the strong biradical character of dis_TS. For example in 2007 Piecuch et al. showed that even CCSD(T) with cc-pVTZ basis sometimes considered as the golden standard of quantum chemistry dramatically underestimates the activation barrier.7 They obtained 21.8 kcal/ mol which lies significantly below the barrier for con_TS. For the latter CCSD(T) yields 40.4 kcal/mol in very good agreement with the MRCI, MRPT, and experimental results. Accordingly, CCSD(T) gives a wrong description of the present isomerization reaction since it favors the disrotatoric pathway. In the past decade Piecuch et al. developed completely renormalized coupled cluster methods, namely, CR-CCSD(T)8 * To whom correspondence should be addressed, luechow@ rwth-aachen.de.
and CR-CC(2,3)9 which turned out to be suitable for describing multireference systems. In 2007 and 2008 they revealed that the CR-CC approaches correctly place the disrotatory pathway above the conrotatory one and the CR-CC activation energies corresponding to the conrotatory pathway, particularly the CRCC(2,3) one, are in excellent agreement with experiment.7,10 The failures described previously are often encountered when nondynamical correlation comes into play or more precisely when energy differences between species which comprise different portions of nondynamical correlation are to be calculated. A single reference ansatz like CCSD(T) provides the major portion of dynamical correlation energy. In comparison the amount of the nondynamical correlation energy is smaller and so there is no effective error cancellation. On the other hand the description of biradical species can be improved by applying multireference methods like multiconfigurational or complete active space self-consistent field (MCSCF or CASSCF), MRCI and various types of MRPT methods. The major drawback is again associated with the error cancellation between open and closed shell systems. Since the latter methods are designed to account for strong nondynamical effects, they fail in giving an accurate description of strong dynamical correlation that dominate closed shell systems. With respect to the latter, one can state that only computational methods which accurately balance dynamical and nondynamical correlation effects can be applied for giving a proper description of the present reaction system. Since the fixed-node DMC method (FN-DMC) always accounts for both dynamical and nondynamical correlation effects, it is considered as a suitable method. The DMC results are about to be compared with those obtained by exploiting the CR-CC methods mentioned above. Method The FN-DMC has been discussed elaborately in the literature and will be discussed here only very briefly.11,12 The method is based on the formal solution of the Schro¨dinger equation in imaginary time (τ ) it)
∂Ψ(x, τ) ˆ Ψ(x, τ) ) 1 ∇2Ψ(x, τ) - V(x)Ψ(x, τ) ) -H ∂τ 2
10.1021/jp108605g 2010 American Chemical Society Published on Web 12/01/2010
(1)
Isomerization of Bicyclo[1.1.0]butane
J. Phys. Chem. A, Vol. 114, No. 50, 2010 13223
SCHEME 1: Schematic Reaction Scheme of the Isomerizationa
a
The detailed geometries employed for the calculation presented in this paper have been taken from ref 7.
which is a generalized diffusion equation. The solution is ˆ obtained by applying the operator e-Hτ to an arbitrary initial function Ψ(x, 0) ˆ
Ψ(x, τ) ) e-HτΨ(x, 0)
(2)
For large τ this propagator equation yields the ground state solution of equation eq 1. With regard to a more efficient algorithm a guide function ΦG is introduced. In a Monte Carlo simulation the stochastic solution of the underlying differential equation is expressed in terms of an ensemble of random walkers {Y(i)} where a random walker contains the spatial coordinates of the electrons. The resulting random walk process is governed by a drift-diffusion step of the following form (i) (i) Yn+1 ) Y(i) n + b(Yn )∆τ + ∆W · √∆τ
ΦG ) eU
U)
) ] (4)
with the local energy13
ˆ ΦG H ΦG
(5)
After a certain equilibration time, the energy is calculated as the weighted average of the local energy
E(FN) ) 0
∑ W(Y(i))EL(Y(i)) i
∑ W(Y(i))
∑ UIij I,i
